On partially τ-quasinormal subgroups of finite groups

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1 Hacettepe Journal of Mathematics and Statistics Volume 43 (6) (2014), On partially τ-quasinormal subgroups of finite groups Changwen Li, Xuemei Zhang and Xiaolan Yi Abstract Let H be a subgroup of a group G. We say that: (1) H is τ-quasinormal in G if H permutes with every Sylow subgroup Q of G such that ( H, Q ) = 1 and ( H, Q G ) 1; (2) H is partially τ-quasinormal in G if G has a normal subgroup T such that HT is S-quasinormal in G and H T H τg, where H τg is the subgroup generated by all those subgroups of H which are τ-quasinormal in G. In this paper, we find a condition under which every chief factor of G below a normal subgroup E of G is cyclic by using the partial τ-quasinormality of some subgroups AMS Classification: 20D10, 20D20. Keywords: S-quasinormal, partially τ-quasinormal, cyclic. Received 05 : 06 : 2013 : Accepted 03 : 10 : 2013 Doi : /HJMS Introduction All groups considered in the paper are finite. The notations and terminology in this paper are standard, as in [4] and [6]. G always denotes a finite group, π(g) denotes the set of all prime dividing G and F (G) is the generalized Fitting subgroup of G, i.e., the product of all normal quasinilpotent subgroups of G. Normal subgroup plays an important role in the study of the structure of groups. Many authors are interested to extend the concept of normal subgroup. For example, a subgroup H of G is said to be S-quasinormal [7] in G if H permutes with every Sylow subgroup of G. As a generalization of S-quasinormality, a subgroup H of G is said to be τ-quasinormal [11] in G if H permutes with every Sylow subgroup Q of G such that ( H, Q ) = 1 and ( H, Q G ) 1. On the other hand, Wang [17] extended normality as School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou, , China lcw2000@126.com Department of Basic Sciences, Yancheng Institute of Technology, Yancheng, , China School of Science, Zhejiang Sci-Tech University, Hangzhou, , China The project is supported by the Natural Science Foundation of China (No: ) and the Priority Academic Program Development of Jiangsu Higher Education Institutions.

2 follows: a subgroup H of G is said to be c-normal in G if there exists a normal subgroup K of G such that HK = G and H K H G, where H G is the maximal normal subgroup of G contained in H. In the literature, many people have studied the influence of the τ-quasinormality and c-normality on the structure of finite groups and obtained many interesting results (see [2, 5, 8, 11, 12, 17, 19]). As a development, we now introduce a new concept: 1.1. Definition. A subgroup H of a group G is said to be partially τ-quasinormal in G if there exists a normal subgroup T of G such that HT is S-quasinormal in G and H T H τg, where H τg is the subgroup generated by all those subgroups of H which are τ-quasinormal in G. Clearly, partially τ-quasinormal subgroup covers both the concepts of τ-quasinormal subgroup and c-normal subgroup. However, the following examples show that the converse is not true Example. Let G = S 4 be the symmetric group of degree 4. (1) Let H be a Sylow 3-subgroup of G and N the normal abelian 2-subgroup of G of order 4. Then HN = A 4 G and H N=1. Hence H is a partially τ-quasinormal subgroup of G. But, obviously, H is not c-normal in G. (2) Let H= (14). Obviously, HA 4=G and H A 4=1. Hence H is partially τ- quasinormal in G. But, obviously, H is not τ-quasinormal in G. A normal subgroup E of a group G is said to be hypercyclically embedded in G if every chief factor of G below E is cyclic. The product of all normal hypercyclically embedded subgroups of G is denoted by Z U (G). In [15] and [16], Skiba gave some characterizations of normal hypercyclically embedded subgroups related to S-quasinormal subgroups. The main purpose of this paper is to give a new characterization by using partially τ-quasinormal property of maximal subgroups of some Sylow subgroups. We obtain the following result. Main Theorem. Let E be a normal subgroup of G. Suppose that there exists a normal subgroup X of G such that F (E) X E and X satisfies the following properties: for every non-cyclic Sylow p-subgroup P of X, every maximal subgroup of P not having a supersoluble supplement in G is partially τ-quasinormal in G. Then E is hypercyclically embedded in G. The following theorems are the main stages in the proof of Main Theorem Theorem. Let P be a Sylow p-subgroup of a group G, where p is a prime divisor of G with ( G, p 1) = 1. If every maximal subgroup of P not having a p-nilpotent supplement in G is partially τ-quasinormal in G, then G is soluble Theorem. Let P be a Sylow p-subgroup of a group G, where p is a prime divisor of G with ( G, p 1) = 1. Then G is p-nilpotent if and only if every maximal subgroup of P not having a p-nilpotent supplement in G is partially τ-quasinormal in G Theorem. Let E be a normal subgroup in G and let P be a Sylow p-subgroup of E, where p is a prime divisor of E with ( E, p 1) = 1. Suppose that every maximal subgroup of P not having a p-supersoluble supplement in G is partially τ-quasinormal in G. Then each chief factor of G between E and O p (E) is cyclic Theorem. Let E be a normal subgroup of a group G. Suppose that for each p π(e), every maximal subgroup of non-cyclic Sylow p-subgroup P of E not having a p-supersoluble supplement in G is partially τ-quasinormal in G. Then every chief factor of G below E is cyclic.

3 2. Preliminaries 2.1. Lemma ( [3] and [7]). Suppose that H is a subgroup of G and H is S-quasinormal in G. Then (1) If H K G, then H is S-quasinormal in K. (2) If N is a normal subgroup of G, then HN is S-quasinormal in G and HN/N is S-quasinormal in G/N. (3) If K G, then H K is S-quasinormal in K. (4) H is subnormal in G. (5) If K G and K is S-quasinormal in G, then H K is S-quasinormal in G Lemma ([11, Lemmas 2.2 and 2.3]). Let G be a group and H K G. (1) If H is τ-quasinormal in G, then H is τ-quasinormal in K. (2) Suppose that H is normal in G and π(k/h) = π(k). If K is τ-quasinormal in G, then K/H is τ-quasinormal in G/H. (3) Suppose that H is normal in G. Then EH/H is τ-quasinormal in G/H for every τ-quasinormal subgroup E in G satisfying ( H, E ) = 1. (4) If H is τ-quasinormal in G and H O p(g) for some prime p, then H is S- quasinormal in G. (5) H τg H τk. (6) Suppose that K is a p-group and H is normal in G. Then K τg/h (K/H) τ(g/h). (7) Suppose that H is normal in G. Then E τgh/h (EH/H) τ(g/h) for every p-subgroup E of G satisfying ( H, E ) = Lemma. Let G be a group and H G. Then (1) If H is partially τ-quasinormal in G and H K G, then H is partially τ- quasinormal in K. (2) Suppose that N G and N H. If H is a p-group and H is partially τ- quasinormal in G, then H/N is partially τ-quasinormal in G/N. (3) Suppose that H is a p-subgroup of G and N is a normal p -subgroup of G. If H is partially τ-quasinormal in G, then HN/N is partially τ-quasinormal in G/N. (4) If H is partially τ-quasinormal in G and H K G, then there exists T G such that HT is S-quasinormal in G, H T H τg and HT K. Proof. (1) Let N be a normal subgroup of G such that HN is S-quasinormal in G and H N H τg. Then K N K, H(K N) = HN K is S-quasinormal in K by Lemma 2.1(3) and H (K N) = H N H τg H τk by Lemma 2.2(5). Hence H is partially τ-quasinormal in K. (2) Suppose that H is partially τ-quasinormal in G. Then there exists K G such that HK is S-quasinormal in G and H K H τg. This implies that KN/N G/N and (H/N)(KN/N) = HK/N is S-quasinormal in G/N by Lemma 2.1(2). In view of Lemma 2.2(6), H/N KN/N = (H K)N/N H τgn/n = H τg/n (H/N) τ(g/n). Thus H/N is partially τ-quasinormal in G/N. (3) Suppose that H is partially τ-quasinormal in G. Then there exists K G such that HK is S-quasinormal in G and H K H τg. Clearly, KN/N G and (HN/N)(KN/N) = HKN/N is S-quasinormal in G/N by Lemma 2.1(2). On the other hand, since ( HN : H, HN : N )=1, HN/N KN/N = (HN K)N/N = (H K)(N K)N/N = (H K)N/N H τgn/n. In view of Lemma 2.2(7), we have H τgn/n (HN/N) τ(g/n). Hence HN/N is partially τ-quasinormal in G/N. (4) Suppose that H is partially τ-quasinormal in G. Then there exists N G such that HN is S-quasinormal in G and H N H τg. Let T = N K. Then T G, HT = H(N K) = HN K is S-quasinormal in G by Lemma 2.1(5), HT K and H T = H N K = H N H τg.

4 2.4. Lemma. Let G be a group and p a prime dividing G with ( G, p 1) = 1. (1) If N is normal in G of order p, then N lies in Z(G). (2) If G has cyclic Sylow p-subgroups, then G is p-nilpotent. (3) If M G and G : M = p, then M G. (4) If G is p-supersoluble, then G is p-nilpotent. Proof. (1), (2) and (3) can be found in [18, Theorem 2.8]. Now we only prove (4). Let A/B be an arbitrary chief factor of G. If G is p-supersolvable, then A/B is either a cyclic group with order p or a p -group. If A/B = p, then Aut(A/B) = p 1. Since G/C G(A/B) is isomorphic to a subgroup of Aut((A/B), the order of G/C G(A/B) must divide ( G, p 1) = 1, which shows that G = C G(A/B). Therefore, we have G is p-nilpotent Lemma ([10, Lemma 2.12]). Let P be a Sylow p-subgroup of a group G, where p is a prime divisor of G with ( G, p 1) = 1. If every maximal subgroup of P has a p-nilpotent supplement in G, then G is p-nilpotent Lemma ([13, Theorem A]). If P is an S-quasinormal p-subgroup of a group G for some prime p, then N G(P ) O p (G) Lemma ([6, VI, 4.10]). Assume that A and B are two subgroups of a group G and G AB. If AB g = B g A holds for any g G, then either A or B is contained in a nontrivial normal subgroup of G Lemma ([20, Chap.1, Theorem 7.19]). Let H be a normal subgroup of G. Then H Z U (G) if and only if H/Φ(H) Z U (G/Φ(H)) Lemma ([14, Lemma 2.11]). Let N be an elementary abelian normal subgroup of a group G. Assume that N has a subgroup D such that 1 < D < N and every subgroup H of N satisfying H = D is S-quasinormal in G. Then some maximal subgroup of N is normal in G Lemma. Let N be a non-identity normal p-subgroup of a group G. If N is elementary and every maximal subgroup of N is partially τ-quasinormal in G, then some maximal subgroup of N is normal in G. Proof. If N = p, then it is clear. Let L be a non-identity minimal normal p-subgroup of G contained in N. First we assume that N L. By Lemma 2.3(2), the hypothesis still holds on G/L. Then by induction some maximal subgroup M/L of N/L is normal in G/L. Clearly, M is a maximal subgroup of N and M is normal in G. Consequently the lemma follows. Now suppose that L = N. Let M be any maximal subgroup of N. Then by the hypothesis, there exists T G such that MT is S-quasinormal in G and M T M τg. Suppose that M M τg. Then MT M and T 1. If N MT, then N = N MT = M(N T ). Hence N T, which implies that M = M T = M τg, a contradiction. If N MT, then M = M(T N) = MT N is S-quasinormal in G by Lemma 2.1(5), a contradiction again. Hence M = M τg. In view of Lemma 2.2(4), M is S-quasinormal in G. By Lemma 2.9, some maximal subgroup of N is normal in G. Thus the lemma holds Lemma ([15, Theorem B]). Let F be any formation and G a group. If H G and F (H) Z F (G), then H Z F (G).

5 3. Proofs of Theorems Proof of Theorem 1.3. Assume that this theorem is false and let G be a counterexample with minimal order. We proceed the proof via the following steps. (1) O p(g) = 1. Assume that L = O p(g) 1. Clearly, P/L is a Sylow p-subgroup of G/L. Let M/L be a maximal subgroup of P/L. Then M is a maximal subgroup of P. If M has a p-nilpotent supplement D in G, then M/L has a p-nilpotent supplement DL/L in G/L. If M is partially τ-quasinormal in G, then M/L is partially τ-quasinormal in G/L by Lemma 2.3(2). Hence G/L satisfies the hypothesis of the theorem. The minimal choice of G implies that G/L is soluble. Consequently, G is soluble. This contradiction shows that step (1) holds. (2) O p (G) = 1. Assume that R = O p (G) 1. Then, obviously, P R/R is a Sylow p-subgroup of G/R. Suppose that M/R is a maximal subgroup of P R/R. Then there exists a maximal subgroup P 1 of P such that M = P 1R. If P 1 has a p-nilpotent supplement D in G, then M/R has a p-nilpotent supplement DR/R in G/R. If P 1 is partially τ-quasinormal in G, then M/R is partially τ-quasinormal in G/R by Lemma 2.3(3). The minimal choice of G implies that G/R is soluble. By the well known Feit-Thompson s theorem, we know that R is soluble. It follows that G is soluble, a contradiction. (3) P is not cyclic. If P is cyclic, then G is p-nilpotent by Lemma 2.4, and so G is soluble, a contradiction. (4) If N is a minimal normal subgroup of G, then N is not soluble. Moreover, G = P N. If N is p-soluble, then O p(n) 1 or O p (N) 1. Since O p(n) char N G, O p(n) O p(g). Analogously O p (N) O p (G). Hence O p(g) 1 or O p (G) 1, which contradicts step (1) or step (2). Therefore N is not soluble. Assume that P N < G. By Lemma 2.3(1), every maximal subgroup of P not having a p-nilpotent supplement in P N is partially τ-quasinormal in P N. Thus P N satisfies the hypothesis. By the minimal choice of G, P N is soluble and so N is soluble. This contradiction shows that G = P N. (5) G has a unique minimal normal subgroup N. By step (4), we see that G = P N for every normal subgroup N of G. It follows that G/N is soluble. Since the class of all soluble groups is closed under subdirect product, G has a unique minimal normal subgroup, say N. (6) The final contradiction. If every maximal subgroup of P has a p-nilpotent supplement in G, then, in view of Lemma 2.5, G is p-nilpotent and so G is soluble. This contradiction shows that we may choose a maximal subgroup P 1 of P such that P 1 is partially τ-quasinormal in G. Then there exists a normal subgroup T of G such that P 1T is S-quasinormal in G and P 1 T (P 1) τg. If T = 1, then P 1 is S-quasinormal in G. In view of Lemma 2.6, P 1 P O p (G) = G. By step (5), P 1 = 1 or N P 1. Since N is not soluble by step (4), we have that P 1 = 1. Consequently, P is cyclic, which contradicts step (3). Hence T 1 and N T. It follows that P 1 N = (P 1) τg N. For any Sylow q-subgroup N q of N with q p, N q is also a Sylow q-subgroup of G by step (4). From step (2) it is easy to see that (P 1) τgn q = N q(p 1) τg. Then (P 1) τgn q N = N q((p 1) τg N) = N q(p 1 N), i.e., P 1 N is τ-quasinormal in N. Since N is a direct product of some isomorphic non-abelian simple groups, we may assume that N = N 1 N k. By Lemma 2.2(1), P 1 N is τ-quasinormal in (P 1 N)N 1. Thus (P 1 N)(N 1q) n 1 N 1 = (N 1q) n 1 (P 1 N N 1) = (N 1q) n 1 (P 1 N 1) for any n 1 N 1, where N 1q is a Sylow q-subgroup of N 1 with q p. Since (N 1q) n 1 (P 1 N 1) N 1, we have N 1 is not simple by Lemma 2.7, a contradiction.

6 Proof of Theorem 1.4. If G is p-nilpotent, then G has a normal Hall p -subgroup G p. Let P 1 be any maximal subgroup of P. Then G : P 1G p = p. In view of Lemma 2.4(3), P 1G p G. Obviously, P 1 G p = 1. Hence P 1 is partially τ-quasinormal in G. Now we prove the sufficient part. Assume that the assertion is false and let G be a counterexample with minimal order. (1) G is soluble. It follows directly from Theorem 1.3. (2) G has a unique minimal normal subgroup N such that G/N is p-nilpotent. Moreover, Φ(G) = 1. Let N be a minimal normal subgroup of G. Since G is solvable by step (1), N is an elementary abelian subgroup. It is easy to see that G/N satisfies the hypothesis of our theorem by Lemma 2.3. By the minimal choice of G, G/N is p-nilpotent. Since the class of all p-nilpotent groups is a saturated formation, N is a unique minimal normal subgroup of G and Φ(G) = 1. (3) P is not cyclic. If P is cyclic, G is p-nilpotent by Lemma 2.4(2), a contradiction. (4) O p (G) = 1. (5) Every maximal subgroup of P has a p-nilpotent supplement in G. It is clear that N O p(g). By Φ(G) = 1, we may choose a maximal subgroup M of G such that G = NM and G/N = M. Let P 1 be an arbitrary maximal subgroup of P. We will show P 1 has a p-nilpotent supplement in G. Since N has the p-nilpotent supplement M in G, we only need to prove N P 1 when P 1 is partially τ-quasinormal in G. Let T be a normal subgroup of G such that P 1T is S-quasinormal in G and P 1 T (P 1) τg. First, we assume that T = 1, i.e., P 1 is S-quasinormal in G. In view of Lemma 2.6, P 1 P O p (G) = G. By virtue of Lemma 2.4(2) and step (3), P 1 1. Hence N P 1 by step (2). Now, assume that T 1. Then N T. It follows that P 1 N = (P 1) τg N. For any Sylow q-subgroup G q of G (p q), (P 1) τgg q = G q(p 1) τg in view of step (4). Then (P 1) τg N = (P 1) τgg q N (P 1) τgg q. Obviously, P 1 N P. Therefore P 1 N is normal in G. By the minimality of N, we have P 1 N = N or P 1 N = 1. If the later holds, then the order of N is p since P 1 N is a maximal subgroup of N. Consequently, G is p-nilpotent by step (2) and Lemma 2.4(1). This contradiction shows that P 1 N = N and so N P 1. (6) The final contradiction. Since every maximal subgroup of P has a p-nilpotent supplement in G by step (5), we have G is p-nilpotent by Lemma 2.5, a contradiction. Proof of Theorem 1.5. Assume that this theorem is false and and consider a counterexample (G, E) for which G E is minimal. (1) E is p-nilpotent. Let P 1 be a maximal subgroup of P. If P 1 has a p-supersolvable supplement T in G, then P 1 has a p-supersolvable supplement T E in E. Since ( E, p 1) = 1, T E is also p-nilpotent by Lemma 2.4(4). If P 1 is partially τ-quasinormal in G, then P 1 is also partially τ-quasinormal in E by Lemma 2.3(1). Hence every maximal subgroup of P not having a p-nilpotent supplement in E is partially τ-quasinormal in E. In view of Theorem 1.4, E is p-nilpotent. (2) P = E. By step (1), O p (E) is the normal Hall p -subgroup of E. Suppose that O p (E) 1. It is easy to see that the hypothesis of the theorem holds for (G/O p (E), E/O p (E)). By induction, every chief factor of G/O p (E) between E/O p (E) and 1 is cyclic. Consequently, each chief factor of G between E and O p (E) is cyclic. This condition shows that O p (E) = 1 and so P = E.

7 (3) Φ(P ) = 1. Suppose that Φ(P ) 1. By Lemma 2.3(2), it is easy to see that the hypothesis of the theorem holds for (G/Φ(P ), P/Φ(P )). By the choice of (G, E), every chief factor of G/Φ(P ) below P/Φ(P ) is cyclic. In view of Lemma 2.8, every chief factor of G below P is cyclic, a contradiction. (4) Every maximal subgroup of P is partially τ-quasinormal in G. Suppose that there is some maximal subgroup V of P such that V has a p-supersolvable supplement B in G, then G = P B and P B 1. Since P B B, we may assume that B has a minimal normal subgroup N contained in P B. It is clear that N = p. Since P is elementary abelian and G = P B, we have that N is also normal in G. It is easy to see that the hypothesis is still true for (G/N, P/N). Hence every chief factor of G/N below P/N is cyclic by virtue of the choice of (G, E). It follows that every chief factor of G below P is cyclic. This contradiction shows that all maximal subgroups of P are partially τ-quasinormal in G. (5) P is not a minimal normal subgroup of G. Suppose that P is a minimal normal subgroup of G, then some maximal subgroup of P is normal in G by Lemma 2.10, which contradicts the minimality of P. (6) If N is a minimal normal subgroup of G contained in P, then P/N Z U (G/N), N is the only minimal normal subgroup of G contained in P and N > p. Indeed, by Lemma 2.3(2), the hypothesis holds on (G/N, P/N) for any minimal normal subgroup N of G contained in P. Hence every chief factor of G/N below P/N is cyclic by the choice of (G, E) = (G, P ). If N = p, every chief factor of G below P is cyclic, a contradiction. If G has two minimal normal subgroups R and N contained in P, then NR/R P/R and from the G-isomorphism NR/R = N we have N = p, a contradiction. Hence, (6) holds. (7) The final contradiction. Let N be a minimal normal subgroup of G contained in P and N 1 any maximal subgroup of N. We show that N 1 is S-quasinormal in G. Since P is an elementary abelian p-group, we may assume that D is a complement of N in P. Let V = N 1D. Obviously, V is a maximal subgroup of P. By step (4), V is partially τ-quasinormal in G. By Lemma 2.3(4), there exist a normal subgroup T of G such that V T is S- quasinormal in G, V T V τg and V T P. In view of Lemma 2.2(4), V τg is an S-quasinormal subgroup of G. If T = P, then V = V τg is S-quasinormal in G and hence V N = N 1D N = N 1(D N) = N 1 is S-quasinormal in G by Lemma 2.1(5). If T = 1, then V = V T is S-quasinormal in G. Consequently, we have also N 1 is S- quasinormal in G. Now we assume that 1 < T < P. Hence N T by step (6). Then, N 1 = V N = V τg N is S-quasinormal in G by virtue of Lemma 2.1(5). Hence some maximal subgroup of N is normal in G by Lemma 2.9. Consequently, N = p. This contradicts step (6). Proof of Theorem 1.6. Let q be the smallest prime dividing E. In view of step (1) of the proof of Theorem 1.5, E is q-nilpotent. Let E q be the normal Hall q -subgroup of E. If E q = 1, then every chief factor of G below E is cyclic by Theorem 1.5. Hence we may assume that E q 1. Since E q char E G, we see that E q G. By Lemma 2.3(3), the hypothesis of the theorem holds for (G/E q, E/E q ). By induction, every chief factor of G/E q below E/E q is cyclic. On the other hand, (G, E q ) also satisfies the hypothesis of the theorem in view of Lemma 2.3(1). By induction again, we have also every chief factor of G below E q is cyclic. Hence it follows that every chief factor of G below E is cyclic.

8 Proof of Main Theorem. Applying Theorem 1.6, X is hypercyclically embedded in G. Since F (E) X, we have that F (E) is also hypercyclically embedded in G. By virtue of Lemma 2.11, E is also hypercyclically embedded in G. 4. Some Applications 4.1. Theorem. Let F be a saturated formation containing U and E a normal subgroup of a group G such that G/E F. Suppose that for every non-cyclic Sylow subgroup P of E, every maximal subgroup of P not having a supersoluble supplement in G is partially τ-quasinormal in G. Then G F. Proof. Applying our Main Theorem, every chief factor of G below E is cyclic. Since F contains U, we know E is contained in the F -hypercentre of G. From G/E F, it follows that G F Theorem. Let F be a saturated formation containing U and E a normal subgroup of a group G such that G/E F. Suppose that for every non-cyclic Sylow subgroup P of F (E), every maximal subgroup of P not having a supersoluble supplement in G is partially τ-quasinormal in G. Then G F. Proof. The proof is similar to that of Theorem Corollary ([9, Theorem 3.4]). Let F be a saturated formation containing U and E a normal subgroup of a group G such that G/E F. If every maximal subgroup of any Sylow subgroup of F (E) is S-quasinormal in G, then G F Corollary ([19, Theorem 3.4]). Let F be a saturated formation containing U and E a normal subgroup of a group G such that G/E F. If every maximal subgroup of any Sylow subgroup of F (E) is c-normal in G, then G F Corollary ([1, Theorem 1.4]). Let F be a saturated formation containing U and E a soluble normal subgroup of a group G such that G/E F. If every maximal subgroup of any Sylow subgroup of F (E) is S-quasinormal in G, then G F Corollary ([8, Theorem 2]). Let G be a group and E a soluble normal subgroup of G such that G/E is supersolvable. If all maximal subgroups of the Sylow subgroups of F (E) are c-normal in G, then G is supersolvable. References [1] M. Asaad, On maximal subgroups of Sylow subgroups of finite groups, Comm. Algebra, 26(1998), [2] A. Ballester-Bolinches and Y. Wang, Finite groups with some c-normal minimal subgroups, J. Pure Appl. Algebra, 153 (2000), [3] W. E. Deskins, On quasinormal subgroups of finite groups, Math. Z. 82 (1963), [4] W. Guo, The Theory of Class of Groups, Science Press-Kluwer Academic Publishers, Beijing- New York-Dordrecht-Boston-London, [5] X. Guo and K. P. Shum, On c-normal maximal and minimal subgroups of Sylow p-subgroups of finite groups, Arch. Math. 80 (2003), [6] B. Huppert, Endliche Gruppen I, Springer-Verlag, Berlin-Heidelberg-New York, [7] O. H. Kegel, Sylow Gruppen und subnormalteiler endlicher Gruppen, Math. Z. 78 (1962), [8] D. Li and X. Guo, The influence of c-normality of subgroups on the structure of finite groups II, Comm. Algebra, 26 (1998), [9] Y. Li, H. Wei and Y. Wang, The influence of π-quasinormality of some subgroups of a finite griup, Arch. Math. 81 (2003),

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